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<div class="section">
<div class="titlepage"><div><div><h3 class="title">
<a name="boost_icl.concepts.map_traits"></a><a class="link" href="map_traits.html" title="Map Traits">Map Traits</a>
</h3></div></div></div>
<p>
        Icl maps differ in their behavior dependent on how they handle <a href="http://en.wikipedia.org/wiki/Identity_element" target="_top"><span class="emphasis"><em><span class="bold"><strong>identity elements</strong></span></em></span></a> of the associated
        type <code class="computeroutput"><span class="identifier">CodomainT</span></code>.
      </p>
<h5>
<a name="boost_icl.concepts.map_traits.h0"></a>
        <span class="phrase"><a name="boost_icl.concepts.map_traits.remarks_on_identity_elements"></a></span><a class="link" href="map_traits.html#boost_icl.concepts.map_traits.remarks_on_identity_elements">Remarks
        on Identity Elements</a>
      </h5>
<p>
        In the pseudo code snippets below <code class="computeroutput"><span class="number">0</span></code>
        will be used to denote <a href="http://en.wikipedia.org/wiki/Identity_element" target="_top"><code class="computeroutput"><span class="identifier">identity</span> <span class="identifier">elements</span></code></a>,
        which can be different objects like <code class="computeroutput"><span class="keyword">const</span>
        <span class="keyword">double</span> <span class="number">0.0</span></code>,
        empty sets, empty strings, null-vectors etc. dependent of the instance type
        for parameter <code class="computeroutput"><span class="identifier">CodomainT</span></code>.
        The existence of an <span class="emphasis"><em><span class="bold"><strong>identity element</strong></span></em></span>
        wrt. an <code class="computeroutput"><span class="keyword">operator</span><span class="special">+=</span></code>
        is a requirement for template type <code class="computeroutput"><span class="identifier">CodomainT</span></code>.
      </p>
<div class="informaltable"><table class="table">
<colgroup>
<col>
<col>
<col>
</colgroup>
<thead><tr>
<th>
                <p>
                  type
                </p>
              </th>
<th>
                <p>
                  operation
                </p>
              </th>
<th>
                <p>
                  identity element
                </p>
              </th>
</tr></thead>
<tbody>
<tr>
<td>
                <p>
                  <code class="computeroutput"><span class="keyword">int</span></code>
                </p>
              </td>
<td>
                <p>
                  addition
                </p>
              </td>
<td>
                <p>
                  <code class="computeroutput"><span class="number">0</span></code>
                </p>
              </td>
</tr>
<tr>
<td>
                <p>
                  <code class="computeroutput"><span class="identifier">string</span></code>
                </p>
              </td>
<td>
                <p>
                  concatenation
                </p>
              </td>
<td>
                <p>
                  <code class="computeroutput"><span class="string">""</span></code>
                </p>
              </td>
</tr>
<tr>
<td>
                <p>
                  <code class="computeroutput"><span class="identifier">set</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;</span></code>
                </p>
              </td>
<td>
                <p>
                  union
                </p>
              </td>
<td>
                <p>
                  <code class="computeroutput"><span class="special">{}</span></code>
                </p>
              </td>
</tr>
</tbody>
</table></div>
<p>
        In these cases the <code class="computeroutput"><span class="identifier">identity</span> <span class="identifier">element</span></code> value is delivered by the default
        constructor of the maps <code class="computeroutput"><span class="identifier">CodomainT</span></code>
        type. But there are well known exceptions like e.g. numeric multiplication:
      </p>
<div class="informaltable"><table class="table">
<colgroup>
<col>
<col>
<col>
</colgroup>
<thead><tr>
<th>
                <p>
                  type
                </p>
              </th>
<th>
                <p>
                  operation
                </p>
              </th>
<th>
                <p>
                  identity element
                </p>
              </th>
</tr></thead>
<tbody><tr>
<td>
                <p>
                  <code class="computeroutput"><span class="keyword">int</span></code>
                </p>
              </td>
<td>
                <p>
                  multiplication
                </p>
              </td>
<td>
                <p>
                  <code class="computeroutput"><span class="number">1</span></code>
                </p>
              </td>
</tr></tbody>
</table></div>
<p>
        Therefore icl functors, that serve as <code class="computeroutput"><span class="identifier">Combiner</span></code>
        parameters of icl Maps implement a static function <code class="computeroutput"><span class="identifier">identity_element</span><span class="special">()</span></code> to make sure that the correct <code class="computeroutput"><span class="identifier">identity_element</span><span class="special">()</span></code>
        is used in the implementation of <span class="emphasis"><em>aggregate on overlap</em></span>.
</p>
<pre class="programlisting"><span class="identifier">inplace_times</span><span class="special">&lt;</span><span class="keyword">int</span><span class="special">&gt;::</span><span class="identifier">identity_element</span><span class="special">()</span> <span class="special">==</span> <span class="number">1</span>
<span class="comment">// or more general</span>
<span class="identifier">inplace_times</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;::</span><span class="identifier">identity_element</span><span class="special">()</span> <span class="special">==</span> <span class="identifier">unit_element</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;::</span><span class="identifier">value</span><span class="special">()</span>
</pre>
<p>
      </p>
<h5>
<a name="boost_icl.concepts.map_traits.h1"></a>
        <span class="phrase"><a name="boost_icl.concepts.map_traits.definedness_and_storage_of_identity_elements"></a></span><a class="link" href="map_traits.html#boost_icl.concepts.map_traits.definedness_and_storage_of_identity_elements">Definedness
        and Storage of Identity Elements</a>
      </h5>
<p>
        There are two <span class="emphasis"><em>properties</em></span> or <span class="emphasis"><em>traits</em></span>
        of icl maps that can be chosen by a template parameter <code class="computeroutput"><span class="identifier">Traits</span></code>.
        The <span class="emphasis"><em><span class="bold"><strong>first trait</strong></span></em></span> relates
        to the <span class="emphasis"><em><span class="bold"><strong>definedness</strong></span></em></span>
        of the map. Icl maps can be <span class="bold"><strong>partial</strong></span> or
        <span class="bold"><strong>total</strong></span> on the set of values given by domain
        type <code class="computeroutput"><span class="identifier">DomainT</span></code>.
      </p>
<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
<li class="listitem">
            A <span class="emphasis"><em><span class="bold"><strong>partial</strong></span></em></span> map is
            only defined on those key elements that have been inserted into the Map.
            This is usually expected and so <span class="emphasis"><em><span class="bold"><strong>partial
            definedness</strong></span></em></span> is the default.
          </li>
<li class="listitem">
            Alternatively an icl Map can be <span class="emphasis"><em><span class="bold"><strong>total</strong></span></em></span>.
            It is then considered to contain a <span class="emphasis"><em><span class="bold"><strong>neutral
            value</strong></span></em></span> for all key values that are not stored in
            the map.
          </li>
</ul></div>
<p>
        The <span class="emphasis"><em><span class="bold"><strong>second trait</strong></span></em></span> is
        related to the representation of <code class="computeroutput"><span class="identifier">identity</span>
        <span class="identifier">elements</span></code> in the map. An icl map
        can be a <span class="emphasis"><em><span class="bold"><strong>identity absorber</strong></span></em></span>
        or a <span class="emphasis"><em><span class="bold"><strong>identity enricher</strong></span></em></span>.
      </p>
<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
<li class="listitem">
            A <span class="emphasis"><em><span class="bold"><strong>identity absorber</strong></span></em></span>
            never stores value pairs <code class="computeroutput"><span class="special">(</span><span class="identifier">k</span><span class="special">,</span><span class="number">0</span><span class="special">)</span></code> that
            carry identity elements.
          </li>
<li class="listitem">
            A <span class="emphasis"><em><span class="bold"><strong>identity enricher</strong></span></em></span>
            stores value pairs <code class="computeroutput"><span class="special">(</span><span class="identifier">k</span><span class="special">,</span><span class="number">0</span><span class="special">)</span></code>.
          </li>
</ul></div>
<p>
        For the template parameter <code class="computeroutput"><span class="identifier">Traits</span></code>
        of icl Maps we have the following four values.
      </p>
<div class="informaltable"><table class="table">
<colgroup>
<col>
<col>
<col>
</colgroup>
<thead><tr>
<th>
              </th>
<th>
                <p>
                  identity absorber
                </p>
              </th>
<th>
                <p>
                  identity enricher
                </p>
              </th>
</tr></thead>
<tbody>
<tr>
<td>
                <p>
                  partial
                </p>
              </td>
<td>
                <p>
                  partial_absorber <span class="emphasis"><em>(default)</em></span>
                </p>
              </td>
<td>
                <p>
                  partial_enricher
                </p>
              </td>
</tr>
<tr>
<td>
                <p>
                  total
                </p>
              </td>
<td>
                <p>
                  total_absorber
                </p>
              </td>
<td>
                <p>
                  total_enricher
                </p>
              </td>
</tr>
</tbody>
</table></div>
<h5>
<a name="boost_icl.concepts.map_traits.h2"></a>
        <span class="phrase"><a name="boost_icl.concepts.map_traits.map_traits_motivated"></a></span><a class="link" href="map_traits.html#boost_icl.concepts.map_traits.map_traits_motivated">Map
        Traits motivated</a>
      </h5>
<p>
        Map traits are a late extension to the <span class="bold"><strong>icl</strong></span>.
        Interval maps have been used for a couple of years in a variety of applications
        at Cortex Software GmbH with an implementation that resembled the default
        trait. Only the deeper analysis of the icl's <span class="emphasis"><em><span class="bold"><strong>aggregating
        Map's concept</strong></span></em></span> in the course of preparation of the library
        for boost led to the introduction of map Traits.
      </p>
<h6>
<a name="boost_icl.concepts.map_traits.h3"></a>
        <span class="phrase"><a name="boost_icl.concepts.map_traits.add_subtract_antinomy_in_aggregating_maps"></a></span><a class="link" href="map_traits.html#boost_icl.concepts.map_traits.add_subtract_antinomy_in_aggregating_maps">Add-Subtract
        Antinomy in Aggregating Maps</a>
      </h6>
<p>
        Constitutional for the absorber/enricher propery is a little antinomy.
      </p>
<p>
        We can insert value pairs to the map by <span class="emphasis"><em><span class="bold"><strong>adding</strong></span></em></span>
        them to the map via operations <code class="computeroutput"><span class="identifier">add</span><span class="special">,</span> <span class="special">+=</span></code> or <code class="computeroutput"><span class="special">+</span></code>:
</p>
<pre class="programlisting"><span class="special">{}</span> <span class="special">+</span> <span class="special">{(</span><span class="identifier">k</span><span class="special">,</span><span class="number">1</span><span class="special">)}</span> <span class="special">==</span> <span class="special">{(</span><span class="identifier">k</span><span class="special">,</span><span class="number">1</span><span class="special">)}</span> <span class="comment">// addition</span></pre>
<p>
      </p>
<p>
        Further addition on common keys triggers aggregation:
</p>
<pre class="programlisting"><span class="special">{(</span><span class="identifier">k</span><span class="special">,</span><span class="number">1</span><span class="special">)}</span> <span class="special">+</span> <span class="special">{(</span><span class="identifier">k</span><span class="special">,</span><span class="number">1</span><span class="special">)}</span> <span class="special">==</span> <span class="special">{(</span><span class="identifier">k</span><span class="special">,</span><span class="number">2</span><span class="special">)}</span> <span class="comment">// aggregation for common key k</span></pre>
<p>
      </p>
<p>
        A subtraction of existing pairs
</p>
<pre class="programlisting"><span class="special">{(</span><span class="identifier">k</span><span class="special">,</span><span class="number">2</span><span class="special">)}</span> <span class="special">-</span> <span class="special">{(</span><span class="identifier">k</span><span class="special">,</span><span class="number">2</span><span class="special">)}</span> <span class="special">==</span> <span class="special">{(</span><span class="identifier">k</span><span class="special">,</span><span class="number">0</span><span class="special">)}</span> <span class="comment">// aggregation for common key k</span></pre>
<p>
        yields value pairs that are associated with 0-values or <code class="computeroutput"><span class="identifier">identity</span>
        <span class="identifier">elements</span></code>.
      </p>
<p>
        So once a value pair is created for a key <code class="computeroutput"><span class="identifier">k</span></code>
        it can not be removed from the map via subtraction (<code class="computeroutput"><span class="identifier">subtract</span><span class="special">,</span> <span class="special">-=</span></code> or <code class="computeroutput"><span class="special">-</span></code>).
      </p>
<p>
        The very basic fact on sets, that we can remove what we have previously added
</p>
<pre class="programlisting"><span class="identifier">x</span> <span class="special">-</span> <span class="identifier">x</span> <span class="special">=</span> <span class="special">{}</span></pre>
<p>
        does not apply.
      </p>
<p>
        This is the motivation for the <span class="emphasis"><em><span class="bold"><strong>identity absorber</strong></span></em></span>
        Trait. A identity absorber map handles value pairs that carry identity elements
        as <span class="emphasis"><em><span class="bold"><strong>non-existent</strong></span></em></span>, which
        saves the law:
</p>
<pre class="programlisting"><span class="identifier">x</span> <span class="special">-</span> <span class="identifier">x</span> <span class="special">=</span> <span class="special">{}</span></pre>
<p>
      </p>
<p>
        Yet this introduces a new problem: With such a <span class="emphasis"><em>identity absorber</em></span>
        we are <span class="emphasis"><em>by definition</em></span> unable to store a value <code class="computeroutput"><span class="special">(</span><span class="identifier">k</span><span class="special">,</span><span class="number">0</span><span class="special">)</span></code> in the map.
        This may be unfavorable because it is not inline with the behavior of stl::maps
        and this is not necessarily expected by clients of the library.
      </p>
<p>
        The solution to the problem is the introduction of the identity enricher
        Trait, so the user can choose a map variant according to her needs.
      </p>
<h6>
<a name="boost_icl.concepts.map_traits.h4"></a>
        <span class="phrase"><a name="boost_icl.concepts.map_traits.partial_and_total_maps"></a></span><a class="link" href="map_traits.html#boost_icl.concepts.map_traits.partial_and_total_maps">Partial and
        Total Maps</a>
      </h6>
<p>
        The idea of a identity absorbing map is, that an <span class="emphasis"><em><span class="bold"><strong>associated
        identity element</strong></span></em></span> value of a pair <code class="computeroutput"><span class="special">(</span><span class="identifier">k</span><span class="special">,</span><span class="number">0</span><span class="special">)</span></code> <span class="emphasis"><em><span class="bold"><strong>codes non-existence</strong></span></em></span>
        for it's key <code class="computeroutput"><span class="identifier">k</span></code>. So the pair
        <code class="computeroutput"><span class="special">(</span><span class="identifier">k</span><span class="special">,</span><span class="number">0</span><span class="special">)</span></code>
        immediately tunnels from a map where it may emerge into the realm of non
        existence.
</p>
<pre class="programlisting"><span class="special">{(</span><span class="identifier">k</span><span class="special">,</span><span class="number">0</span><span class="special">)}</span> <span class="special">==</span> <span class="special">{}</span></pre>
<p>
      </p>
<p>
        If identity elements do not code <span class="emphasis"><em><span class="bold"><strong>non-existence</strong></span></em></span>
        but <span class="emphasis"><em><span class="bold"><strong>existence with null quantification</strong></span></em></span>,
        we can also think of a map that has an associated identity element <span class="emphasis"><em><span class="bold"><strong>for every</strong></span></em></span> key <code class="computeroutput"><span class="identifier">k</span></code>
        that has no associated value different from 0. So in contrast to modelling
        <span class="bold"><strong>all</strong></span> neutral value pairs <code class="computeroutput"><span class="special">(</span><span class="identifier">k</span><span class="special">,</span><span class="number">0</span><span class="special">)</span></code> as being <span class="emphasis"><em><span class="bold"><strong>non-existent</strong></span></em></span>
        we can model <span class="bold"><strong>all</strong></span> neutral value pairs <code class="computeroutput"><span class="special">(</span><span class="identifier">k</span><span class="special">,</span><span class="number">0</span><span class="special">)</span></code> as being
        <span class="emphasis"><em><span class="bold"><strong>implicitly existent</strong></span></em></span>.
      </p>
<p>
        A map that is modelled in this way, is one large vector with a value <code class="computeroutput"><span class="identifier">v</span></code> for every key <code class="computeroutput"><span class="identifier">k</span></code>
        of it's domain type <code class="computeroutput"><span class="identifier">DomainT</span></code>.
        But only non-identity values are actually stored. This is the motivation
        for the definedness-Trait on <code class="computeroutput"><span class="identifier">icl</span>
        <span class="identifier">Maps</span></code>.
      </p>
<p>
        A <span class="emphasis"><em><span class="bold"><strong>partial</strong></span></em></span> map models
        the intuitive view that only value pairs are existent, that are stored in
        the map. A <span class="emphasis"><em><span class="bold"><strong>total</strong></span></em></span> map
        exploits the possibility that all value pairs that are not stored can be
        considered as being existent and <span class="emphasis"><em><span class="bold"><strong>quantified</strong></span></em></span>
        with the identity element.
      </p>
<h5>
<a name="boost_icl.concepts.map_traits.h5"></a>
        <span class="phrase"><a name="boost_icl.concepts.map_traits.pragmatical_aspects_of_map_traits"></a></span><a class="link" href="map_traits.html#boost_icl.concepts.map_traits.pragmatical_aspects_of_map_traits">Pragmatical
        Aspects of Map Traits</a>
      </h5>
<p>
        From a pragmatic perspective value pairs that carry <code class="computeroutput"><span class="identifier">identity</span>
        <span class="identifier">elements</span></code> as mapped values can often
        be ignored. If we count, for instance, the number of overlaps of inserted
        intervals in an <code class="computeroutput"><a class="link" href="../../boost/icl/interval_map.html" title="Class template interval_map">interval_map</a></code>
        (see example <a class="link" href="../examples/overlap_counter.html" title="Overlap counter">overlap counter</a>),
        most of the time, we are not interested in whether an overlap has been counted
        <code class="computeroutput"><span class="number">0</span></code> times or has not been counted
        at all. A identity enricher map is only needed, if we want to distinct between
        non-existence and 0-quantification.
      </p>
<p>
        The following distinction can <span class="bold"><strong>not</strong></span> be made
        for a <code class="computeroutput"><a class="link" href="../../boost/icl/partial_absorber.html" title="Struct partial_absorber">partial_absorber</a></code>
        map but it can be made for an <code class="computeroutput"><a class="link" href="../../boost/icl/partial_enricher.html" title="Struct partial_enricher">partial_enricher</a></code>
        map:
      </p>
<pre class="programlisting">(k,v) does not exist in the map: Pair (k,v) has NOT been dealt with
(k,0) key k carries 0          : Pair (k,v) has     been dealt with resulting in v=0
</pre>
<p>
        Sometimes this subtle distinction is needed. Then a <code class="computeroutput"><a class="link" href="../../boost/icl/partial_enricher.html" title="Struct partial_enricher">partial_enricher</a></code>
        is the right choice. Also, If we want to give two <code class="computeroutput"><span class="identifier">icl</span><span class="special">::</span><span class="identifier">Maps</span></code>
        a common set of keys in order to, say, iterate synchronously over both maps,
        we need <span class="emphasis"><em>enrichers</em></span>.
      </p>
</div>
<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
<td align="left"></td>
<td align="right"><div class="copyright-footer">Copyright © 2007-2010 Joachim
      Faulhaber<br>Copyright © 1999-2006 Cortex Software
      GmbH<p>
        Distributed under the Boost Software License, Version 1.0. (See accompanying
        file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
      </p>
</div></td>
</tr></table>
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